LOYOLA COLLEGE (AUTONOMOUS), CHENNAI – 600 034
M.Sc. DEGREE EXAMINATION – STATISTICS
THIRD SEMESTER – April 2009
YB 42
ST 3809 - STOCHASTIC PROCESSES
Date & Time: 23/04/2009 / 1:00 - 4:00 Dept. No.
Max. : 100 Marks
PART-A
Answer all the questions:
(10 X 2 = 20)
1. Define a stochastic process with an example.
2. Define a process with independent increments.
3. Show that communication between two states i and j satisfies transitive relation.
4. Define (i) transcient state (ii) recurrent state.
5. Define a Markov process.
6.
Obtain the PGF of a Poisson process.
7. Define a renewal function. What is the relation between a renewal function and the
distribution functions of inter occurrence times?
8. When do you say that X n  is a martingale with respect to Yn  ?
9. What is a branching process?
10. What is the relationship between Poisson process and exponential distribution?
PART-B
Answer any 5 questions:
(5 X 8 = 40)
11. State and prove Chapman – Kolmogorov equation for a discrete time Markov chain.
12. Obtain the equation for Pn t  in a Yule process with X(0) = 1.
13. Let Y0  0 and Yi , i  1.2,3,....... be i.i.d random variables with mean 0 and variance  2 .
2
 n 
Show that X n    Yi   n 2 is a martingale with respect to Yn .
 i 1 
14. Show that the matrix of transition probabilities together with the initial distribution
completely specifies a Markov chain.
15. Show that the renewal function satisfies
t
M t   F t    M t  y dF  y 
0
1
16. Establish the relationship between Poisson process and Binomial distribution.
17. Obtain the stationary distribution for the Markov chain having transition probability
matrix
 0 2 / 3 1 / 3
1 / 2 0 1 / 2


1 / 2 1 / 2 0 
18. If a process has stationary independent increments and finite mean show that
EX t   m0  m1t where m0  EX 0  and m1  EX 1   m0 .
PART-C
Answer any 2 questions:
(2 X 20 = 40)
19. a) State and prove the necessary and sufficient condition required by a state to be
recurrent.
b.) Verify whether state 0 is recurrent in a symmetric random walk in three dimensions.
(10+10)
20. a) State the postulates of a Poisson process. Obtain the expression for Pn t  .
b.) Obtain the distribution for waiting time of k arrivals for a Poisson process.
(15+5)
21. a) Obtain the generating function for a branching process. Hence obtain the mean and
variance.
b) Let Pk , k  0,1,2 be the probability that an individual in a generation generates k
1
1
1
off springs. If P0  , P1  , P2  obtain the probability of extinction.
4
4
2
(15+5)
22. a.) Obtain the renewal function corresponding to the lifetime density.
f x   2 xex , x  0,   0
b.) Show that the likelihood ratio forms a martingale.
c.) Let X n  be a martingale with respect to Yn . If  is a convex function with


E   X n    show that   X n  is a sub martingale with respect to Yn .
(10+5+5)

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